Understanding and Predicting Bubble Growth in Bubbling Fluidized Beds via Physics-Aware Data-Driven Models

Bubble dynamics in bubbling fluidized bed reactors govern heat and mass transfer rates, mixing uniformity, and overall process efficiency, but remain challenging to predict accurately. This study develops a hybrid modeling framework to model bubble growth along bed height by integrating empirical correlations with data-driven corrections using Physics-Informed Neural Networks (PINNs) and Universal Differential Equations (UDEs). Experimental data used for training, which were obtained for Geldart Group B particles with various particle size distributions (PSDs), are characteristically noisy with non-normal bubble diameter distributions. The optimal black-box data-driven NN fails to capture the known monotonic bubble growth with bed height. In contrast, the PINN, embedding the Hilligardt–Werther empirical correlation, enforces physical monotonicity and robustness despite the noisy data. The UDE further couples this empirical correlation with a learned neural correction, yielding a physically interpretable, data-calibrated model. Results show that the UDE preserves the physical trend while adaptively compensating for empirical correlation discrepancies, with the data-driven neural correction contributing 20–35% of the total trend. The largest data-driven correction occurs for the broadest PSD case, which is expected because such conditions likely lie outside the calibration range of the empirical correlation. Importantly, this also demonstrates the ability of the hybrid framework to extend empirical models beyond their original scope. In the absence of first-principles-based descriptions, the hybrid (gray-box) approach presented here reconciles physical underpinnings with data-driven flexibility, offering a more reliable, interpretable, and generalizable framework for modeling bubble growth in practical fluidized beds.